Such a system might be too complex for a human to consistently/safely implement, but I am curious if they exist. I am imagining a system where your choice of suit or rank to lead or follow with could convey additional information mid-hand.
For a starter, it is obvious that such signalling only applies on defence - Declarer has no need to communicate with his partner (Dummy), and such communication is anyways illegal. Let's start with the basic defensive systems and progress from there. All of these are perfectly legal when fully explained to the opponents (except some of the more complex techniques are restricted in limited-master-point events).
You will of course notice that many of these are contradictory or ambiguous - that is the cost of asking a very restricted language to convey a rich amount of information. Established partnerships discuss the priority of these signals at great length, but the general gist is (usually): Attitude first; Count once attitude is known; Suit-preference once count is known, or when obligatory. Deciphering that consistently with your partner is one of the most highly rewarded skills in Bridge, as most of your opponents are so very bad at it.
More complex signals involve adding a mixture of these variations to the basic scheme:
That's all I can recall off the top of my head, but I have seen attempts to play even more unusual combinations. Myself, I don't think the benefit is worth the risk of a disaster from misunderstanding.
Update: With some exceptions, any partnership agreement on the meaning of a legal call made or card played, or sequence of such, that is well explained in advance of the round/match to the opponents is legal. The exceptions generally cover methods deemed too complex, in the context of the competition being held, for the opponents to defend and understand. Generally there are few restrictions in 10-day World Team Championships, and very many in 3 hours novice match-point games.
The section on Carding on each chart of current ACBL Convention Charts explains the defensive carding systems allowed or disallowed for the corresponding class of competitions held under the aegis of the American Contract Bridge League.
Update #2 - Partnership Agreements:
In addition there is PRISM signals which with some bidding understanding can give the shape of your partner's hand from his plays. Its somewhat complicated.
Cards played constitute signals, and there are several basic types. For the partner of the leader, they include the following.
Attitude signals refer to playing a high spot card on a lead to show that you like the suit.
Count signals were better discussed in Pieter's post. Briefly, one example is playing high-low in a no trump contract to show an even number of cards, and low-high to show an odd number.
Suit preference signals are made when leading back a suit that is not a trump suit. Besides the main suit and the trump suit, there are two side suits. A high preference lead means "I have an entry in the higher side suit." A low lead for preference means, "I have an entry in the lower side suit." One expert suggested using an intermediate card to signal neither a preference in the high or low side suit, meaning that trumps should be led.
For the leader, there are several "conventions." The most common ones are "top of a sequence" if you have a sequence of at least three, or three out of four, and "fourth best," from a long suit in which you have no sequence. "Top of nothing" is a common "passive" lead when the leader is "playing not to lose" (tricks).